Betz Type Limitation of Vortex Wind Machines

This paper describes the theoretical power limitations of four vortex type wind machines. The maximum possible power output is presented in the form of a Betz-type dimensionless power coefficient. All these vortex wind machines use a vortex generator in the form of either a low aspect ratio wing or vanes in a tower. These parts do not rotate but have to move with the wind direction and therefore are non-stationary; consequently their projected area has been selected for the reference area. All four machines are proven here to have a power coefficient below that of an actuator disk in a free stream. Three machines use a turbine rotor placed coaxially with the core of a vortex. The first two machines have a vortex generated upstream of the rotor and ingest the viscous part of the vortex. The total pressure deficit in the viscous core of the vortex reduces their power output in proportion to the square of the vortex strength. The second two machines have a vortex generated downstream of the rotor and use the viscous core as a vaccum pump to drive the wind turbine. The vortex generating device is much larger than the turbine, especially for the second pair of machines, which further lowers the obtainable power coefficient.